Hermitian positive definite matrix

[math8]

Let P and Q be Hermitian positive definite matrices.
We prove that x*Px < or eq. x*Qx, for all x in C^n (C : complex numbers) if and only if x*Q^-1 x < or eq. x*P^-1 x for all x in C^n.

I guess I should use the definition of a hermitian positive definite matrix being
x*Px > 0 , for all x in C^n but I am not sure how to proceed to get both the P and Q in the inequality.

Should I try and multiply both sides of the inequality by x and x*?

 

[jbunniii]

Try proving it in the special case where P and Q are also diagonal. Then for the general case, use the fact that you can write

$$P = A^{-1}D_P A$$
$$Q = B^{-1}D_Q B$$

where $$D_P$$ and $$D_Q$$ are both diagonal. (This is the spectral theorem.)